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In the proof of complete reducibility in my Lie Algebra lecture notes the following fact appears ($\rho$ is a finite dimensional representation, $V$ a complex vector space and $\mathfrak{g}$ is complex semisimple):

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Now why can we assume that $\mathfrak{g} \subset \mathfrak{gl}(V)$?

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  • $\begingroup$ Is $\rho$ faithful? If it's semisimple you can find a faithful representation because the killing form is not degenerate, but if $\rho$ is a specific representation I don't know... Anyway look at Ado's Theorem, you can always assume to be in $\mathfrak{gl}(V)$ if your algebra is finite dimensional $\endgroup$ – Dac0 Feb 21 '17 at 14:23
  • $\begingroup$ It has nothing to do with Ado's theorem. The point is that a representation is semisimple iff the obvious (faithful) representation of its image is semisimple. This is essentially a tautology. $\endgroup$ – YCor Feb 22 '17 at 7:00

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